Cycle lengths in graphs of given minimum degree

TL;DR

The paper strengthens stability results for cycle-length distributions in graphs with given minimum degree. It proves that for $2$-connected graphs with $\delta\ge k\ge4$, one obtains $k$ admissible cycles or, modulo-$k$ cycle families, all residues (with sharp extremal exceptions $K_{k+1}$ and $K_{k,n-k}$), and an even-$k$ refinement under $\delta\ge k-1$. It then extends these modular-cycle results to all residues in the non-bipartite setting, while providing a Turán-type bound for the maximum number of edges avoiding a $0$ modulo $k$ cycle, with tight cases described. Collectively, the work offers a comprehensive stability analysis that generalizes and sharpens prior results of Gao–Huo–Liu–Ma and connects cycle-length phenomena to extremal graph theory.

Abstract

In a graph, $k$ cycles are {\em admissible} if their lengths form an arithmetic progression with common difference one or two. Let $G$ be a 2-connected graph with minimum degree at least $k\geqslant 4$. We prove that \begin{itemize} \item [(1)] $G$ contains $k$ admissible cycles, unless $G\cong K_{k+1}$ or $K_{k,n-k}$; \item [(2)] $G$ contains cycles of lengths $\ell$ modulo $k$ for all even $\ell$, unless $G\cong K_{k+1}$ or $K_{k,n-k}$; \item [(3)] $G$ contains cycles of lengths $\ell$ modulo $k$ for all $\ell$, unless $G\cong K_{k+1}$ or $G$ is bipartite. \end{itemize} In addition, we show that if $k$ is even and $G$ is 2-connected with minimum degree at least $k-1$ and order at least $k+2$, then $G$ contains cycles of lengths $\ell$ modulo $k$ for all even $\ell$. These findings provide a stability analysis of the main results on cycle lengths in graphs of given minimum degree in [J. Gao, Q. Huo, C. Liu, J. Ma, A unified proof of conjectures on cycle lengths in graphs, International Mathematics Research Notices 2022 (10) (2022) 7615--7653]. As a corollary, we determine the maximum number of edges in a graph that does not contain a cycle of length 0 modulo $k$ for all odd $k$.

Figures (5)

• Figure 1: Constructions of $S$, $T$, $Q$ and $G_i$ for $i\in \{1,2,3,4\}$ in Lemma \ref{['lemma: 2-con-bi-H']}.
• Figure 2: Graph $F_r$.
• Figure 3: Hypo-Petersen graphs.
• Figure 4: The 6-, 7-, 8-, 9-cycles in a hypo-Petersen graph.
• Figure 5: Constructions of $L_i(x,y)$ ($i=1,\ldots,4$) in Lemma \ref{['lemma: two paths differing by 2 mod 4']}.

Theorems & Definitions (139)

• Theorem 1.1: Gao, Huo, Liu and Ma G+22
• Theorem 1.2: Gao, Huo, Liu and Ma G+22
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8: Chen and Saito CS94
• Theorem 1.9: Győri et al. G+23
• Theorem 1.10